IMA Preprint Series # 2202

Transcription

1 FRIENDY EQUIIBRIUM INTS IN EXTENSIVE GMES WITH CMETE INFRMTIN By Ezo Mch IM epnt Sees # My 8 INSTITUTE FR MTHEMTICS ND ITS ICTINS UNIVERSITY F MINNEST nd Hll 7 Chuch Steet S.E. Mnnepols Mnnesot hone: Fx: UR:

2 Fendly equlbum ponts n extensve gmes wth complete nfomton by Ezo Mch * ** bstct: In ths note we pove n exstence theoem egdng fendly equlbum ponts n extensve gmes wth complete nfomton. The fendly equlbum ponts s efnement of equlbum ponts. * Decto del Insttuto de Mtemátc plcd. Unvesdd Nconl de Sn us CNICET Eécto de los ndes 95. C.. 57 Sn us. gentn ** Ths ppe hs been ptlly suppoted by gnt fom the CNICET.

3 Intoducton nd the Fomulton The mpotnt noton of equlbum ponts ws ntoduced by Nsh 8 fo n- peson gmes n noml fom nd ts exstence ws poved n the sme ppe. Howeve the exstence of equlbum ponts n n-peson gmes wth complete nfomton n extensve gmes ws pove n dffeent context whch mght be consulted n the excellent boos by Buge Kühn Myeson 6 nd/o Vn Dmme. In ecent note we hve studed the exstence ponts of E-ponts n n-peson gmes n extensve fom wth complete nfomton. The ede wth nteest mght ed. Thee we hve gven suffcent nd necessy condton fo the exstence of E- ponts whch genelze equlbum ponts. In nothe non publshed ppe Mch 5 ntoduced the noton of fendly equlbum ponts nd poved n exstence theoem unde genel condton. ll ths mtel s povded n noml n-peson gmes. lve n 9 hs extended the fendly equlbum ponts un the context of Gc Judo who consequently extended t ths tme the pope nd pefect equlbum ponts due to Myeson 6 nd Selten. In ths note we e gong to pove genel theoem concenng the exstence of fendly equlbum ponts n extensve gmes wth complete nfomton. Consde n-peson extensve gme wth complete nfomton gven by the set of plyes N { n} nd the chnce plye. The set of the nodes of the ooted tee s G. The oot s. G s pttoned n the sets G N G G G N nd G then The end ponts of the tee e E E. We do not need explctly such ponts. Fo ech g G N we expess by g ll the edges emnng fom g. et t p g g the coespondng ssgned pobblty. complete pln fo plye N s sttegy nmely Now... n X N { g}. g G detemnes dstbuton of pobblty t the end ponts of the tee. Theefoe we hve f t ech end pont we hve the pyoff functon the expectton functon s complete detemned nd the expecttons e clled by buse of lnguge pyoff functons. They e wtten s

4 + n... n whee s s usul n non coopetve theoy of gmes. f...f Now fo ech plye N we consde stng of dffeent plyes f. Then fendly equlbum pont s pont such tht Σ f f f f f f whee { : } Fst we e gong to gve n exmple tht n equlbum pont s not fendly equlbum pont. Consde the smple followng tee b b b b l l l l g g m m Whee {} G {g g }. The pont {l l m } s n G equlbum pont unde the condton of the pyoff functons nd b b. Howeve f nd b > bthe pont s not fendly equlbum pont wth the fendly stuctue f f f. In such cse the pont ~ ~ ~ {l l m } s fendly equlbum pont: ths s esy to see snce t s n equlbum pont nd ~ {l l l l l l l l }

5 %. E M? 9 % F F? F F emnng fom Now ntoducng the notton g G nd fo we wte g g fo the node endng t the edge g g! "$# g the pyoff n the tuncton gme Γ g wth oot g nd &! "$# g. &! "'# g s the estcton of n Next we pesent the esult of ths ppe Theoem: ny n-peson extensve gme wth complete nfomton nd ny fend stuctue hs lwys fendly equlbum pont. oof: We pove t by nducton on the length of the tee. et * be the length of the tee. If λ. Then f G thee s nothng to pove. If G then t s cle tht choosng pont + + Σ σ + + σ f f f σ σ + + f then t s fendly equlbum pont. Such pont s evdent tht exsts. Now we ssume tht fo λ λ Γ. The theoem s tue nd we wll pove tht t s tue fo λ Γ. Consde the oot nd then ll the gmes 6 7$8 buse of notton lso fo. Snce :<; > s λ Γ pncple t hs fendly equlbum pont? Σ p? :; > 9 f. Now f we hve B CD? N nd by then by nducton GH IKJ GH IKJ GH IKJ GH IKJ F Σ p F Σ p M Σ

6 V N W N U U The sme nequltes ppe fo the plyes n the fendly stuctue. Theefoe t s fendly equlbum pont. Now f consde the gme hvng length one ooted by nd whee the pyoffs t the end of f e Then by choosng QR SKT QR ST QR ST such tht fo N QR ST QR SKT f f QR SKT QR SKT f whee the Ω e efeed to ths gme of lenght one then we hve constucted Σ. Now we wll pove tht such pont s fendly equlbum pont n the ente gme. fo Fo plye we fst hve f QR ST QR ST - QR SKT QR ST QR SKT - Q<R ST Q<R ST - W Σ f QR ST QR ST f - QR SKT QR ST QR SKT f - 5

7 _ ^ ^ X ^ X f ^ _ Y<Z [\ Y<Z [\ X f - f... whee V η wth V dsont unon. The fst nequlty n the pevous sequences s due to the fct of choosng the pont the second one s due to the fct tht YZ [\ X. Fo we hve We hve used the fct tht sme hold fo the fend stuctue. competton. YZ [\ X nd s fendly equlbum pont n bc de bc de - bc dke bc de b<c de Σ bc dke Theefoe the theoem s poved. s n equlbum pont n bc de. The In ths wy we hve obtned the poweful tool fo decson mng unde Fnlly we would le to sy tht t s possble to extend the fendly equlbum ponts n extensve gme wth pefect nfomton when the fendly stuctue depends on the node. 6

8 Bblogphy Buge E.: Intoducton to the theoy of gmes. entce Hll 959. Kühn H.: ectue n Gme Theoy. nceton Unvesty ess Gc Judo I.: Nuevos equlbos estbles nte tendencs l eo en uegos no coopetvos. Doctol thess. Unv. Sntgo de Compostel 989. Mch E.: E-ponts n extensve gmes wth complete nfomton to ppe 5 Mch E.: Fendly equlbum ponts to ppe 6 Myeson R.B.: Gme theoy nlyss of Conflct. Hwd Unvesty ess Cmbdge Mss 99 7 Thoms.C.: Gme Theoy nd pplctons. John Wley & Sons 98 8 Nsh J.: Non coopetve gmes. nnls of Mthemtcs 5 pp lve E.: Refnmentos de puntos de equlbo en uegos no coopetvos. Doctol thess. Unv. Nconl de Sn us. Selten R.: Reexmnton of the pefectness concept fo equlbum ponts n extensve gmes. Int. J. Gme Theoy pp Vn Dmme E.: Stblty nd pefecton of Nsh equlb. Spnge Velg Beln 987 7